Prove That a Set Is Uncountably Infinite

Date: 10/31/97 at 00:21:17
From: Matt Ramos
Subject: Prove that a set is uncountably infinite
How can one prove that the set [0,1]x[0,1] is uncountably infinite?
(We are given the hint to show that if this set were put into
one-to-one correspondence with the natural numbers, then so could
[0,1].)
I am having problems picturing the contents of the set and how to set
up a correspondence.
Thank you in advance,
Matt Ramos

Date: 10/31/97 at 05:51:42
From: Doctor Mitteldorf
Subject: Re: Prove that a set is uncountably infinite
Dear Matt,
Picture [0,1] as the set of all points on a line between 0 and 1,
inclusive. The points are so "close together" that you can't count
them.
Your picture for [0,1]x[0,1] is all the points in the square with
corners at (0,0), (0,1), (1,0) and (1,1).
Your intuition should tell you that the number of points in a plane
area must be "greater than" the number of points on a line.
To prove that this set isn't countable, we assume that it is countable
and derive a contradiction. What does it mean to say it's countable?
It just means that there's some systematic method for listing them
all. For example, suppose the list looked like this:
1) (0,0)
2) (0,.1)
3) (.1,0)
4) (.33333..., 0)
5) (.2, .1415926...)
etc...
In fact, there's no rhyme or reason to the above list. There's no
way of telling what the point number (6) is.
But the proof says, suppose there were a method to this madness, and I
claimed to be sure that, by this method, I would cover every single
pair of real numbers between 0 and 1 somewhere in my list.
Well, then you could just take apart my list, throwing out duplicates.
You could construct your own list
1) 0
2) .1
3) .33333...
4) .2
5) .1415926...
that came from just extracting both numbers from my list of ordered
pairs and removing the duplicates, and then you'd have a listing of
the points on a line.
But you proved in class that it's impossible to make a list of points
on the line, so it must have been impossible for me to come up with my
list. QED.
-Doctor Mitteldorf, The Math Forum
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